3.7.31 \(\int \frac {1}{x (c+a^2 c x^2)^2 \arctan (a x)^3} \, dx\) [631]

3.7.31.1 Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=-\frac {1}{2 a c^2 x \arctan (a x)^2}+\frac {a x}{2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)^2}+\frac {1-a^2 x^2}{2 c^2 \left (1+a^2 x^2\right ) \arctan (a x)}+\frac {\text {Si}(2 \arctan (a x))}{c^2}-\frac {\text {Int}\left (\frac {1}{x^2 \arctan (a x)^2},x\right )}{2 a c^2} \]

output

-1/2/a/c^2/x/arctan(a*x)^2+1/2*a*x/c^2/(a^2*x^2+1)/arctan(a*x)^2+1/2*(-a^2 
*x^2+1)/c^2/(a^2*x^2+1)/arctan(a*x)+Si(2*arctan(a*x))/c^2-1/2*Unintegrable 
(1/x^2/arctan(a*x)^2,x)/a/c^2
 

3.7.31.2 Mathematica [N/A]

Not integrable

Time = 1.43 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx \]

input

Integrate[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]
 

output

Integrate[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^3), x]
 

3.7.31.3 Rubi [N/A]

Not integrable

Time = 0.84 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {5501, 27, 5461, 5377, 5467, 5505, 4906, 27, 3042, 3780}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{x \arctan (a x)^3 \left (a^2 c x^2+c\right )^2} \, dx\)

\(\Big \downarrow \) 5501

\(\displaystyle \frac {\int \frac {1}{c x \left (a^2 x^2+1\right ) \arctan (a x)^3}dx}{c}-a^2 \int \frac {x}{c^2 \left (a^2 x^2+1\right )^2 \arctan (a x)^3}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {1}{x \left (a^2 x^2+1\right ) \arctan (a x)^3}dx}{c^2}-\frac {a^2 \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^3}dx}{c^2}\)

\(\Big \downarrow \) 5461

\(\displaystyle \frac {-\frac {\int \frac {1}{x^2 \arctan (a x)^2}dx}{2 a}-\frac {1}{2 a x \arctan (a x)^2}}{c^2}-\frac {a^2 \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^3}dx}{c^2}\)

\(\Big \downarrow \) 5377

\(\displaystyle \frac {-\frac {\int \frac {1}{x^2 \arctan (a x)^2}dx}{2 a}-\frac {1}{2 a x \arctan (a x)^2}}{c^2}-\frac {a^2 \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)^3}dx}{c^2}\)

\(\Big \downarrow \) 5467

\(\displaystyle \frac {-\frac {\int \frac {1}{x^2 \arctan (a x)^2}dx}{2 a}-\frac {1}{2 a x \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-2 \int \frac {x}{\left (a^2 x^2+1\right )^2 \arctan (a x)}dx-\frac {x}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}\)

\(\Big \downarrow \) 5505

\(\displaystyle \frac {-\frac {\int \frac {1}{x^2 \arctan (a x)^2}dx}{2 a}-\frac {1}{2 a x \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-\frac {2 \int \frac {a x}{\left (a^2 x^2+1\right ) \arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}\)

\(\Big \downarrow \) 4906

\(\displaystyle \frac {-\frac {\int \frac {1}{x^2 \arctan (a x)^2}dx}{2 a}-\frac {1}{2 a x \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-\frac {2 \int \frac {\sin (2 \arctan (a x))}{2 \arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\int \frac {1}{x^2 \arctan (a x)^2}dx}{2 a}-\frac {1}{2 a x \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\int \frac {1}{x^2 \arctan (a x)^2}dx}{2 a}-\frac {1}{2 a x \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-\frac {\int \frac {\sin (2 \arctan (a x))}{\arctan (a x)}d\arctan (a x)}{a^2}-\frac {x}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}\)

\(\Big \downarrow \) 3780

\(\displaystyle \frac {-\frac {\int \frac {1}{x^2 \arctan (a x)^2}dx}{2 a}-\frac {1}{2 a x \arctan (a x)^2}}{c^2}-\frac {a^2 \left (-\frac {\text {Si}(2 \arctan (a x))}{a^2}-\frac {x}{2 a \left (a^2 x^2+1\right ) \arctan (a x)^2}-\frac {1-a^2 x^2}{2 a^2 \left (a^2 x^2+1\right ) \arctan (a x)}\right )}{c^2}\)

input

Int[1/(x*(c + a^2*c*x^2)^2*ArcTan[a*x]^3),x]
 

output

$Aborted
 

3.7.31.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3780

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte 
gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
 

rule 4906

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b 
_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x 
]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG 
tQ[p, 0]
 

rule 5377

Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Sy 
mbol] :> Unintegrable[(d*x)^m*(a + b*ArcTan[c*x^n])^p, x] /; FreeQ[{a, b, c 
, d, m, n, p}, x]
 

rule 5461

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_))/((d_) + (e_ 
.)*(x_)^2), x_Symbol] :> Simp[(f*x)^m*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*( 
p + 1))), x] - Simp[f*(m/(b*c*d*(p + 1)))   Int[(f*x)^(m - 1)*(a + b*ArcTan 
[c*x])^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] 
&& LtQ[p, -1]
 

rule 5467

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*(x_))/((d_) + (e_.)*(x_)^2)^2, 
 x_Symbol] :> Simp[x*((a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)*(d + e*x^2 
))), x] + (-Simp[(1 - c^2*x^2)*((a + b*ArcTan[c*x])^(p + 2)/(b^2*e*(p + 1)* 
(p + 2)*(d + e*x^2))), x] - Simp[4/(b^2*(p + 1)*(p + 2))   Int[x*((a + b*Ar 
cTan[c*x])^(p + 2)/(d + e*x^2)^2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && 
EqQ[e, c^2*d] && LtQ[p, -1] && NeQ[p, -2]
 

rule 5501

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/d   Int[x^m*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c 
*x])^p, x], x] - Simp[e/d   Int[x^(m + 2)*(d + e*x^2)^q*(a + b*ArcTan[c*x]) 
^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ[p, 2* 
q] && LtQ[q, -1] && ILtQ[m, 0] && NeQ[p, -1]
 

rule 5505

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(q_), x_Symbol] :> Simp[d^q/c^(m + 1)   Subst[Int[(a + b*x)^p*(Sin[x]^m/ 
Cos[x]^(m + 2*(q + 1))), x], x, ArcTan[c*x]], x] /; FreeQ[{a, b, c, d, e, p 
}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && ILtQ[m + 2*q + 1, 0] && (IntegerQ[q 
] || GtQ[d, 0])
 

3.7.31.4 Maple [N/A] (verified)

Not integrable

Time = 9.63 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00

\[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right )^{2} \arctan \left (a x \right )^{3}}d x\]

input

int(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^3,x)
 

output

int(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^3,x)
 

3.7.31.5 Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.73 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{3}} \,d x } \]

input

integrate(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^3,x, algorithm="fricas")
 

output

integral(1/((a^4*c^2*x^5 + 2*a^2*c^2*x^3 + c^2*x)*arctan(a*x)^3), x)
 

3.7.31.6 Sympy [N/A]

Not integrable

Time = 1.53 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\frac {\int \frac {1}{a^{4} x^{5} \operatorname {atan}^{3}{\left (a x \right )} + 2 a^{2} x^{3} \operatorname {atan}^{3}{\left (a x \right )} + x \operatorname {atan}^{3}{\left (a x \right )}}\, dx}{c^{2}} \]

input

integrate(1/x/(a**2*c*x**2+c)**2/atan(a*x)**3,x)
 

output

Integral(1/(a**4*x**5*atan(a*x)**3 + 2*a**2*x**3*atan(a*x)**3 + x*atan(a*x 
)**3), x)/c**2
 

3.7.31.7 Maxima [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 141, normalized size of antiderivative = 6.41 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{3}} \,d x } \]

input

integrate(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^3,x, algorithm="maxima")
 

output

1/2*(2*(a^4*c^2*x^4 + a^2*c^2*x^2)*arctan(a*x)^2*integrate((3*a^4*x^4 + 2* 
a^2*x^2 + 1)/((a^6*c^2*x^7 + 2*a^4*c^2*x^5 + a^2*c^2*x^3)*arctan(a*x)), x) 
 - a*x + (3*a^2*x^2 + 1)*arctan(a*x))/((a^4*c^2*x^4 + a^2*c^2*x^2)*arctan( 
a*x)^2)
 

3.7.31.8 Giac [N/A]

Not integrable

Time = 121.36 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.14 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{2} x \arctan \left (a x\right )^{3}} \,d x } \]

input

integrate(1/x/(a^2*c*x^2+c)^2/arctan(a*x)^3,x, algorithm="giac")
 

output

sage0*x
 

3.7.31.9 Mupad [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x \left (c+a^2 c x^2\right )^2 \arctan (a x)^3} \, dx=\int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2} \,d x \]

input

int(1/(x*atan(a*x)^3*(c + a^2*c*x^2)^2),x)
 

output

int(1/(x*atan(a*x)^3*(c + a^2*c*x^2)^2), x)